Malfatti circles
A simple construction of the Malfatti circles was given by Steiner (1826), and many mathematicians have since studied the problem.
He assumed that the solution to this problem was given by three tangent circles within the triangular cross-section of the wedge.
[1] Malfatti's work was popularized for a wider readership in French by Joseph Diaz Gergonne in the first volume of his Annales (1811), with further discussion in the second and tenth.
Lob and Richmond (1930), who went back to the original Italian text, observed that for some triangles a larger area can be achieved by a greedy algorithm that inscribes a single circle of maximal radius within the triangle, inscribes a second circle within one of the three remaining corners of the triangle, the one with the smallest angle, and inscribes a third circle within the largest of the five remaining pieces.
[5][6] Even earlier, the same problem was considered in a 1384 manuscript by Gilio di Cecco da Montepulciano, now in the Municipal Library of Siena, Italy.
[7] Jacob Bernoulli (1744) studied a special case of the problem, for a specific isosceles triangle.
Since the work of Malfatti, there has been a significant amount of work on methods for constructing Malfatti's three tangent circles; Richard K. Guy writes that the literature on the problem is "extensive, widely scattered, and not always aware of itself".
Solutions based on algebraic formulations of the problem include those by C. L. Lehmus (1819), E. C. Catalan (1846), C. Adams (1846, 1849), J. Derousseau (1895), and Andreas Pampuch (1904).
[11] Gatto (2000) and Mazzotti (1998) recount an episode in 19th-century Neapolitan mathematics related to the Malfatti circles.
[12] Although much of the early work on the Malfatti circles used analytic geometry, Steiner (1826) provided the following simple synthetic construction.
Two of these bitangents pass between their circles: one is an angle bisector, and the second is shown as a red dashed line in the figure.
The three bitangents x, y, and z cross the triangle sides at the point of tangency with the third inscribed circle, and may also be found as the reflections of the angle bisectors across the lines connecting pairs of centers of these incircles.
[8] The radius of each of the three Malfatti circles may be determined as a formula involving the three side lengths a, b, and c of the triangle, the inradius r, the semiperimeter
Related formulae may be used to find examples of triangles whose side lengths, inradii, and Malfatti radii are all rational numbers or all integers.
Then the three lines AD, BE, and CF meet in a single triangle center known as the first Ajima–Malfatti point after the contributions of Ajima and Malfatti to the circle problem.
